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Theorem ordtypelem10 7944
Description: Lemma for ordtype 7949. Using ax-rep 4550, exclude the possibility that  O is a proper class and does not enumerate all of 
A. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypotheses
Ref Expression
ordtypelem.1  |-  F  = recs ( G )
ordtypelem.2  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
ordtypelem.3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
ordtypelem.5  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
ordtypelem.6  |-  O  = OrdIso
( R ,  A
)
ordtypelem.7  |-  ( ph  ->  R  We  A )
ordtypelem.8  |-  ( ph  ->  R Se  A )
Assertion
Ref Expression
ordtypelem10  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
Distinct variable groups:    v, u, C    h, j, t, u, v, w, x, z, R    A, h, j, t, u, v, w, x, z    t, O, u, v, x    ph, t, x    h, F, j, t, u, v, w, x, z
Allowed substitution hints:    ph( z, w, v, u, h, j)    C( x, z, w, t, h, j)    T( x, z, w, v, u, t, h, j)    G( x, z, w, v, u, t, h, j)    O( z, w, h, j)

Proof of Theorem ordtypelem10
Dummy variables  b 
c  m are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtypelem.1 . . 3  |-  F  = recs ( G )
2 ordtypelem.2 . . 3  |-  C  =  { w  e.  A  |  A. j  e.  ran  h  j R w }
3 ordtypelem.3 . . 3  |-  G  =  ( h  e.  _V  |->  ( iota_ v  e.  C  A. u  e.  C  -.  u R v ) )
4 ordtypelem.5 . . 3  |-  T  =  { x  e.  On  |  E. t  e.  A  A. z  e.  ( F " x ) z R t }
5 ordtypelem.6 . . 3  |-  O  = OrdIso
( R ,  A
)
6 ordtypelem.7 . . 3  |-  ( ph  ->  R  We  A )
7 ordtypelem.8 . . 3  |-  ( ph  ->  R Se  A )
81, 2, 3, 4, 5, 6, 7ordtypelem8 7942 . 2  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
91, 2, 3, 4, 5, 6, 7ordtypelem4 7938 . . . . 5  |-  ( ph  ->  O : ( T  i^i  dom  F ) --> A )
10 frn 5719 . . . . 5  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  ran  O  C_  A )
119, 10syl 16 . . . 4  |-  ( ph  ->  ran  O  C_  A
)
12 simprl 754 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  A
)
136adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R  We  A
)
147adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  R Se  A )
151, 2, 3, 4, 5, 13, 14ordtypelem8 7942 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  ran  O ) )
16 isof1o 6196 . . . . . . . . . . . . 13  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  ran  O )  ->  O : dom  O -1-1-onto-> ran  O )
17 f1of 5798 . . . . . . . . . . . . 13  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O --> ran  O )
1815, 16, 173syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O --> ran  O )
19 f1of1 5797 . . . . . . . . . . . . . 14  |-  ( O : dom  O -1-1-onto-> ran  O  ->  O : dom  O -1-1-> ran 
O )
2015, 16, 193syl 20 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O : dom  O
-1-1-> ran  O )
21 simpl 455 . . . . . . . . . . . . . . 15  |-  ( ( b  e.  A  /\  -.  b  e.  ran  O )  ->  b  e.  A )
22 seex 4831 . . . . . . . . . . . . . . 15  |-  ( ( R Se  A  /\  b  e.  A )  ->  { c  e.  A  |  c R b }  e.  _V )
237, 21, 22syl2an 475 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  { c  e.  A  |  c R b }  e.  _V )
2411adantr 463 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  A
)
25 rexnal 2902 . . . . . . . . . . . . . . . . . . 19  |-  ( E. m  e.  dom  O  -.  ( O `  m
) R b  <->  -.  A. m  e.  dom  O ( O `
 m ) R b )
261, 2, 3, 4, 5, 6, 7ordtypelem7 7941 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( ( O `  m ) R b  \/  b  e.  ran  O ) )
2726ord 375 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  b  e.  A )  /\  m  e.  dom  O )  -> 
( -.  ( O `
 m ) R b  ->  b  e.  ran  O ) )
2827rexlimdva 2946 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  b  e.  A )  ->  ( E. m  e.  dom  O  -.  ( O `  m ) R b  ->  b  e.  ran  O ) )
2925, 28syl5bir 218 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  A. m  e.  dom  O ( O `  m
) R b  -> 
b  e.  ran  O
) )
3029con1d 124 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  A. m  e.  dom  O ( O `  m
) R b ) )
3130impr 617 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. m  e.  dom  O ( O `  m
) R b )
32 ffun 5715 . . . . . . . . . . . . . . . . . . . 20  |-  ( O : ( T  i^i  dom 
F ) --> A  ->  Fun  O )
339, 32syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  Fun  O )
34 funfn 5599 . . . . . . . . . . . . . . . . . . 19  |-  ( Fun 
O  <->  O  Fn  dom  O )
3533, 34sylib 196 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  O  Fn  dom  O
)
3635adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Fn  dom  O )
37 breq1 4442 . . . . . . . . . . . . . . . . . 18  |-  ( c  =  ( O `  m )  ->  (
c R b  <->  ( O `  m ) R b ) )
3837ralrn 6010 . . . . . . . . . . . . . . . . 17  |-  ( O  Fn  dom  O  -> 
( A. c  e. 
ran  O  c R
b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
3936, 38syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ( A. c  e.  ran  O  c R b  <->  A. m  e.  dom  O ( O `  m
) R b ) )
4031, 39mpbird 232 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  A. c  e.  ran  O  c R b )
41 ssrab 3564 . . . . . . . . . . . . . . 15  |-  ( ran 
O  C_  { c  e.  A  |  c R b }  <->  ( ran  O 
C_  A  /\  A. c  e.  ran  O  c R b ) )
4224, 40, 41sylanbrc 662 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  C_  { c  e.  A  |  c R b } )
4323, 42ssexd 4584 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  e.  _V )
44 f1dmex 6743 . . . . . . . . . . . . 13  |-  ( ( O : dom  O -1-1-> ran 
O  /\  ran  O  e. 
_V )  ->  dom  O  e.  _V )
4520, 43, 44syl2anc 659 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  dom  O  e.  _V )
46 fex 6120 . . . . . . . . . . . 12  |-  ( ( O : dom  O --> ran  O  /\  dom  O  e.  _V )  ->  O  e.  _V )
4718, 45, 46syl2anc 659 . . . . . . . . . . 11  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  e.  _V )
481, 2, 3, 4, 5, 13, 14, 47ordtypelem9 7943 . . . . . . . . . 10  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  O  Isom  _E  ,  R  ( dom  O ,  A ) )
49 isof1o 6196 . . . . . . . . . 10  |-  ( O 
Isom  _E  ,  R  ( dom  O ,  A
)  ->  O : dom  O -1-1-onto-> A )
50 f1ofo 5805 . . . . . . . . . 10  |-  ( O : dom  O -1-1-onto-> A  ->  O : dom  O -onto-> A
)
51 forn 5780 . . . . . . . . . 10  |-  ( O : dom  O -onto-> A  ->  ran  O  =  A )
5248, 49, 50, 514syl 21 . . . . . . . . 9  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  ran  O  =  A )
5312, 52eleqtrrd 2545 . . . . . . . 8  |-  ( (
ph  /\  ( b  e.  A  /\  -.  b  e.  ran  O ) )  ->  b  e.  ran  O )
5453expr 613 . . . . . . 7  |-  ( (
ph  /\  b  e.  A )  ->  ( -.  b  e.  ran  O  ->  b  e.  ran  O ) )
5554pm2.18d 111 . . . . . 6  |-  ( (
ph  /\  b  e.  A )  ->  b  e.  ran  O )
5655ex 432 . . . . 5  |-  ( ph  ->  ( b  e.  A  ->  b  e.  ran  O
) )
5756ssrdv 3495 . . . 4  |-  ( ph  ->  A  C_  ran  O )
5811, 57eqssd 3506 . . 3  |-  ( ph  ->  ran  O  =  A )
59 isoeq5 6194 . . 3  |-  ( ran 
O  =  A  -> 
( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
6058, 59syl 16 . 2  |-  ( ph  ->  ( O  Isom  _E  ,  R  ( dom  O ,  ran  O )  <->  O  Isom  _E  ,  R  ( dom 
O ,  A ) ) )
618, 60mpbid 210 1  |-  ( ph  ->  O  Isom  _E  ,  R  ( dom  O ,  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106    i^i cin 3460    C_ wss 3461   class class class wbr 4439    |-> cmpt 4497    _E cep 4778   Se wse 4825    We wwe 4826   Oncon0 4867   dom cdm 4988   ran crn 4989   "cima 4991   Fun wfun 5564    Fn wfn 5565   -->wf 5566   -1-1->wf1 5567   -onto->wfo 5568   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571   iota_crio 6231  recscrecs 7033  OrdIsocoi 7926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-recs 7034  df-oi 7927
This theorem is referenced by:  ordtype  7949
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